How to display a image texture only on the outside of a object (cylinder)?
If the object has different surfaces for the inside and outside of the cup, just assign a white material to the inside.
If there is no thickness and you are seeing the backface of the object on the inside use the Geometry &gt; Backfacing information to control the mix between two different shaders.

Read more about the Geometry node here:
https://docs.blender.org/manual/en/latest/render/shader_nodes/input/geometry.html?highlight=backfacing
how to use a texture on each side of the same face
If you are using a solidify modifier read this answer:
Add different textures to front and back of plane with solidify?